I Canonical Quantization

Suppose we have a classical system described by a Lagrangien [itex]\mathscr{L}(x,t)[/itex].
The same system can be described by the Lagrangien [itex]\mathscr{L'}(x,t)=\mathscr{L}(x,t)+\frac{\mathrm{d}F(x,t)}{\mathrm{d}t}[/itex]. where [itex]F(x,t)[/itex] can be any function.

If we now quantize the system by calculating the Hamiltonian and promoting the canonical position and the canonical momentum to operators, we arrive at two different Hamiltonians describing two different quantum systems.

My question is: How do we choose the Lagrangien to arrive at the "right" quantum system?

Suppose we have a classical system described by a Lagrangien [itex]\mathscr{L}(x,t)[/itex].
The same system can be described by the Lagrangien [itex]\mathscr{L'}(x,t)=\mathscr{L}(x,t)+\frac{\mathrm{d}F(x,t)}{\mathrm{d}t}[/itex]. where [itex]F(x,t)[/itex] can be any function.

If we now quantize the system by calculating the Hamiltonian and promoting the canonical position and the canonical momentum to operators, we arrive at two different Hamiltonians describing two different quantum systems.

My question is: How do we choose the Lagrangien to arrive at the "right" quantum system?

Thank you for your answers :)

The correct equivalence relation is [tex]L'(x , \dot{x},t) = L(x,\dot{x},t) + \frac{d}{dt}F(x,t) .[/tex] The important fact here is [itex]\frac{\partial F}{\partial \dot{x}}=0[/itex], i.e., the primed and un-primed Lagrangians describe the same physical system if and only if the function [itex]F[/itex] dose not depend on the generalized velocity [itex]\dot{x}[/itex]. This equivalence is true in classical mechanics as well as quantum mechanics where it modifies the wave-function by a space-time dependent phase, as expected given that [itex]\Psi \sim \exp (\frac{i}{\hbar}\int L dt )[/itex]. To see this, rewrite the above relation as [tex]L' = L + \frac{\partial F}{\partial x} \dot{x} + \frac{\partial F}{\partial t} .[/tex] From this it follows that the primed generalized momentum [itex]p' = \frac{\partial L'}{\partial \dot{x}}[/itex] is related to un-primed one [itex]p = \frac{\partial L}{\partial \dot{x}}[/itex] by [tex]p' = p + \frac{\partial F}{\partial x} .[/tex] Compare this to the behaviour, under gauge transformation, of the generalized momentum of a charged particle in an electromagnetic field. Now, you can calculate the primed Hamiltonian from [tex]H'\left(x, p'\right) = p' \dot{x} - L' = H \left(x , p' - \frac{\partial F}{\partial x}\right) - \frac{\partial F}{\partial t} .[/tex] Using the above relations, you can easily establish the equivalence between the primed and un-primed Hamilton’s equations as well as the Poisson structure of classical mechanics. Indeed, the transformations [tex]x \to x , \ \ p \to p + \frac{\partial F}{\partial x} ,[/tex] and [tex]H \to H - \frac{\partial F}{\partial t} ,[/tex] are canonical “gauge” transformations.
Let us consider, as simple concrete example, the free particle [itex]L = \frac{1}{2}m \dot{x}^{2}[/itex]. Under a Galilean boost, [itex]\dot{x} \to \dot{x} + \beta[/itex], the Lagrangian [itex]L[/itex] dose not remain invariant. As you can easily see, it changes by total derivative [tex]L \to L' = L + \frac{d}{dt}F(t,x;\beta) , \ \ \ \ \ (1)[/tex][tex]F \equiv m (x \cdot \beta + \frac{1}{2} \beta^{2}t).[/tex]
In classical Mechanics, the system is not affected by the change (1), since the action integral, i.e., the equation of motion remains unchanged. The same is true in QM. Indeed, you can show (though not as easy as the CM case) that, under the Galilean boost, the free particle Schrödinger equation [tex]i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^{2}}{2m}\frac{\partial^{2}\Psi}{\partial x^{2}} ,[/tex] retains its form (in the primed system) if and only if the wave function transforms according to [tex]\Psi (t,x) \to \Psi'(t',x') = \exp \left(\frac{i}{\hbar}F(t,x;\beta) \right) \Psi (t,x) . \ \ \ \ \ \ (2)[/tex] This means that the transformation (1) of the classical system becomes a gauge transformation of the corresponding quantum system. The fact that the same function [itex]F[/itex] appears in (1) as well as (2) means that there is a common cause which can be shown to be the non-trivial cohomology of the Galilei group.

Great posting (put it to the Insights section for reference!). Do you have a citation for where and when Weyl stressed the important fact that pure states are represented by rays rather than vectors in Hilbert space (I guess in his book on group theory in QT which appeared in German in 1931?).

Well, that means that I have to re-type the whole thing again? That will be a waste of my time, for I prefer to produce new post with different material.

Do you have a citation for where and when Weyl stressed the important fact that pure states are represented by rays rather than vectors in Hilbert space (I guess in his book on group theory in QT which appeared in German in 1931?).

Yes, that one as well as "The classical group", Princeton Univ. Press (1946).